# Statistical evaluation of proxies to estimate the erosivity factor of precipitation

### Calculation results and R and PR deviations

The calculation *R*-the factor of the 28 stations varied between 290.8 MJ mm ha^{−1} h^{−1} a^{−1} and 7153.7 MJ mm ha^{−1} h^{−1} a^{−1} with an average of 2424.1 MJ mm ha^{−1} h^{−1} a^{−1}. As shown in Figure 1, all of the proxies examined were found to be surprisingly strongly correlated with the *R*-factor (*r*> 0.62, *p*R -factor in Australia. This finding is at odds with the common notion that regional differences in physiographic parameters preclude a model of the *R* -factor of application to other fields. The reason could be that the *PR* the selected models were established in different regions and developed by the *R*value calculated by the USLE or RUSLE method. The *PR* value is actually an approximation of the *R* -calculation of the USLE or RUSLE factor. Therefore, the *PR* the value is strongly linked to the *R* value, and the difference between *PR* and *R* reflects the correction coefficient, which is the regional difference, and therefore any *PR* values were strongly correlated with *R* values.

The linear regression equation of FIG. 1 can be applied to change the gap between the *PR*and *R*. Table 3 lists the *RMSE* , *ERM* and *NSE* for the estimate of *R* directly used values *PR* and modified *PR.* The direct estimate showed a high discrepancy, the *ERM* ranged from 18.3 to 153.0%, the *RMSE* ranged from 561 to 2279 MJ mm ha^{−1} h^{−1} a^{−1}and the *NSE* ranged from −2.1 to 0.89. Revised calculations, as shown in Table 3, indicated the *ERM* ranged from 13.1 to 77.4%, the *RMSE* ranged from 302 to 1482 MJ mm ha^{−1} h^{−1} a^{−1}and the *NSE* ranged from 0.23 to 0.97. The *RMSE* (*p* = 0.002) and the *ERM*(*p*= 0.01) were significantly reduced, and the *NSE* (*p* = 0.009) increased significantly under the correction, which further proved that the error was chain-reduced when the value changed *PR*has been used. However, all *PR* s showed a similar expression, such as the *RMSE* , *MRE,*and *NSE* of *PR*_{11}, *PR*_{12}, *PR*_{13}and *PR*_{14}which changed slightly after calibration.

### Correlation comparison between R and PR_{S} using Meng’s test

According to the step-by-step testing method devised in this study, the flowchart is shown in Fig. 2, Eq. (9) divided the correlation coefficients by 15 *PR* the sand *R* into two classes: L_{1} (*r*_{1}, *r*_{2}, *r*_{6}, *r*_{9}), and O_{1} (*r*_{3}, *r*_{4}, *r*_{5}, *r*_{seven}, *r*_{8}, *r*_{ten}, *r*_{11}, *r*_{12}, *r*_{13}, *r*_{14}, *r*_{15}), and no significant difference was observed in the L_{1} to classify (*p* > 0.06). The correlation coefficients in the O_{1} class were separated into L_{2} (*r*_{3}, *r*_{4}, *r*_{8}, *r*_{ten}) and O_{2} (*r*_{5}, *r*_{seven}, *r*_{11}, *r*_{12}, *r*_{13}, *r*_{14}, *r*_{15}) classes, and no contrast was detected in the L_{2} to classify (*p*> 0.35). The correlation coefficients in the O_{2} class were divided into L_{3} (*r*_{5}, *r*_{seven}, *r*_{12}) and O_{3} (*r*_{11}, *r*_{13}, *r*_{14}, *r*_{15}) classes, and the coefficients in the L_{3} class showed statistical equality under the Meng test. The correlation coefficients of *PR*_{14} and *PR*_{15} were significantly higher than those of *r*_{11} and *r*_{13,} and other statistical tests indicated no significant difference between *r*_{11} and *r*_{13,} *r*_{14}and *r*_{15}, respectively. After four rounds (Fig. 2), Meng’s stepwise test procedure ranked the correlation between *PR*_{I} and *R*, *r*_{I}as following:

$$r_{1 , } = r_{2} = r_{6} = r_{9}

(14)

As expected, the correlation generally became stronger with finer temporal resolution of precipitation data, and daily proxies were more correlated with *R*than monthly ones, which in turn were more strongly correlated with *R* than annual ones. The *ERM* , *RMSE* and *NSE* (Table 3) showed a very similar rank to the correlation in Eq. (14).

### Recommend PR for rainfall data with different resolutions

Two annual proxies, *PR*_{1} and *PR*_{2} showed a strong correlation with *R*(Fig. 1a,b). The t test indicated that the coefficient of *PR*_{1} was equal to 1, and the constant term of *PR*_{1} was significantly equal to 0. It showed that the *R*the value in Australia can be directly estimated by *PR*_{1}and Table 3 shows that no striking variation was observed among the *ERM*, *RMSE*and *NSE* when estimating the *R* revised value *PR*_{1}. The coefficient and the constant term of *PR*_{2} were significantly different from 1 and 0, respectively. She showed that the estimate of the *R*value in Australia directly by *PR*_{2} would result in a significant discrepancy. Based on eq. (14), no statistical difference was observed between *r*_{1} and *r*_{2}. Therefore, we suggest the use of *PR*_{1} and revised *PR*_{2} as predictors of *R* -factor when only annual rainfall data are available.

Among the monthly proxies, *PR*_{4}, *PR*_{5}and *PR*_{seven} are based on the modified Fournier index (*MFI* ), while *PR*_{6} is based on the Fournier index (*F* ) (Table 1). Both indices have been widely applied to estimate the *R*-factor^{31,32,33,34}. Fox et al.^{ten} and Yue et al. (2014) found that the *MFI* was more strongly correlated with *MAP* that the *F* . Arnoldus^{35} and Hernando and Romana^{36} shows a stronger correlation between the *MFI* and *R* that between the *F* and *R* . The Australian data we used agrees well with the observations above. Although the *MFI*and *F* were both strongly correlated with *MAP*and *R* (Fig. 3), Meng’s test indicated that the *MFI* was more correlated with both *MAP*and *R*(*p*MFI-monthly proxies based on, *PR*_{4}, *PR*_{5}and *PR*_{seven}even showed a stronger correlation with *R* that *PR*_{6} (Eq. (14)), a daily proxy for the *R* -factor. Therefore, we concluded that the *MFI* is higher than the *F* in building a proxy for *R*-factor. The t-test of the regression equation of *PR*_{3}–*PR*_{8} indicated that the coefficients of the equations were significantly different from 1. Table 3 shows that the direct use of the *PR*s to estimate the *R* -factor produced large errors. Therefore, when only monthly rainfall data are available, it is best to use the *PR*_{5} and *PR*_{seven} predict the *R* -factor.

Among the daily proxies, *PR*_{14} and *PR*_{15} were developed in Australia, and their correlation with *R*ranked first among 15 proxies. *PR*_{11} and *PR*_{13}two Chinese proxies, ranked second in terms of correlation with *R*. The reason for the superior performance of *PR*_{14} and *PR*_{15} may not simply be due to the fact that they were based on Australian data. Capturing periodic variation over the course of a year, the model developed by Yu and Rosewell^{37} (the original version of *PR*_{14} and *PR*_{15}) performed best among the 8 *R*-proxies in the northeast of Spain^{seven}.

The t-test of the regression equation of *PR*_{ten}–*PR*_{13} indicates that the coefficient and the constant are equal to 1 (*p*> 0.06) and 0 (*p*> 0.33), respectively. Table 3 shows that no notable improvement was observed in the difference between *R*value when estimated by the revised equations. Therefore, *PR*_{ten}–*PR*_{13} can be used to estimate the *R*assess. The coefficient and the constant term of *PR*_{9} were significantly different from 1 and 0, respectively. The revised version *PR*_{9} was recommended when estimating the *R*-factor in Australia. The coefficient of *PR*_{14} was significantly equal to 1 (*p*= 0.87) and the constant was significantly greater than 0 (*p*= 0.02), indicating that the equation for *PR*_{14} in Fig. 1k probably leads to a large deviation in the estimate *R*values for dry areas. The constant term of *PR*_{15} was equal to 0 (*p*= 0.45) and the coefficient was significantly greater than 1 (*p*= 0.00), indicating that *PR*_{15} underestimated the *R*assess. *PR*_{15} was established using the RUSLE model, which may have led to this result. Nearing et al. (2017) concluded that RUSLE underestimated 14%, compared to RUSLE2 and USLE, in estimating the *R*-factor. This result coincided with the coefficient of 0.19, which is shown in Fig. 1l. We suggest using the revised version *PR*_{14} and *PR*_{15} as substitutes for *R*-factor when daily rainfall data is available. However, with *ERM*s of 26.1% and 23.6%, respectively (Fig. 1i,j), the accuracy remains satisfactory using *PR*_{11} and *PR*_{13} as a substitute for *R*-factor.

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